15.1: *The Time between Successive Generations of Nebulae*

Contrary to common belief it is sometimes easier to talk in mathematics
than to talk in English; this is the reason why many scientific papers
contain more mathematics than is either necessary or desirable. Contrary
to common belief it is also often less precise to do so. For mathematical
symbols have a tendency to conceal the physical meaning that they are
intended to represent; they sometimes serve as a substitute for the arduous
task of deciding what is and what is not relevant; they tend to limit the
inquiry to those physical circumstances that are represented by them and
are by their nature unable to reveal others that may have a decisive
influence on the conclusion that is being sought. It is true that mathematics
cannot lie. But it can mislead.

However, the dangers of over-indulgence in formula spinning are
avoided if mathematics is treated, wherever possible, as a language into
which thoughts may only be translated after they have first been expressed
in the language of words. The use of mathematics in this way is indeed
disciplinary, helpful, and sometimes indispensable.

It is disciplinary in the sense in which it is always disciplinary to
translate a statement from one language into another; this is the best way
of revealing shoddy thinking. It is helpful because some statements are of
such a nature that it is almost, if not quite, impossible to express them in
the language of words; it is reasonable to ask to be spared the ordeal of
attempting to do so. It is indispensable because quantitative conclusions
can only be reached with the help of algebraic expressions.

Such considerations demand now that the comparatively light task be
undertaken of seeking algebraic symbols for some of the concepts that have
been discussed in previous chapters. Only the simplest mathematics is
needed at this stage of the inquiry. But the field of cosmology that is being
studied here is complicated and so far almost unexplored. Once clarity is
achieved concerning its broad outline, the remaining problems will lie
wholly within the mathematician's domain.

It is easy to see that the interval between the moment when a cloud of
one generation begins to form and the moment when one of the next
generation, and in adjoining space, does so must average the time that it
takes for the linear dimensions of space to double themselves. This interval
can be calculated with the help of Hubble's constant *H.*

Let *D* be a given distance in space. The rate at which this increases is

*
dD / dt = HD, *

the solution of which is

*
D = k e*^{Ht}

Let *D = D*_{0} at the moment when one puts * t = 0* and the above expression becomes

*
D = D*_{0} e^{Ht} ...... (15a)

*
H* is, according to a recent estimate, 185 kilometres per second per
megaparsec, which is the same as 5.84 x 10^{19} kilometres per year per
megaparsec and has the dimension of reciprocal time. One megaparsec
is 3.084 x 10^{19} kilometres, from which it follows that

*
H *= 1.89 x 10^{10} reciprocal years.

When the distance doubles one puts *D / D*_{0} = 2 and obtains ln2 = *Ht*, from
which *t* = 3.66 x l0^{9} years.

This means that a new nebula begins to form adjacent to a predecessor
about every three-and-a-half thousand million years.

This is the time during which the lip of a crater on an astronomical
summit goes through the complete evolution of moving outwards, becoming a ridge far from any concentration and flattening sufficiently to attain
at its summit the critical gradient at which a new cloud can begin to
form.

15.2: *The Rate at which an Extragalactic Cloud Grows in Extent*

It is necessary to consider next whether the calculated time that it takes
for the cloud to grow is of the right order of magnitude, lf the cosmological
model based on Symmetrical Impermanence is to resemble actuality, the
cloud must acquire a significant mass during a time that is appreciably
less than 3.66 x 10^{9} years.

For the sake of simplicity let us assume that the cloud is forming on a
pass between two nebulae of equal masses *m*. Although the cloud forms
on a summit and not on a pass, this incorrect assumption will serve to
illustrate the method of calculation and I hope that it will not introduce a
wrong order of magnitude. Let *D*_{c0} be the distance from the pass to one
of the nebulae at the moment when the cloud just begins at the very top
of the pass, and let *D*_{cr} be the distance at a moment when the fringe of
the cloud reaches a distance *r* from the pass. Let E_{c} be the critical potential
gradient at which the rate of origins just balances the rate of loss from
particles falling away down the slope.

We want to find the time that it takes for the fringe of the cloud to reach
a certain fraction of the distance to the nearest galaxy. Let this fraction be
*α *, so that *r = α D*_{cr}

Consider equations (12e) and (12f) . lf (12f) were correct,
*E*_{c} would be reached at distance *r* at the same time as at every other
distance; it would be reached for the separation *D*_{co}. But the correct
equation is (12e) and for this *E*_{c} is not reached until the separation has
become *D*_{cr}. This gives two expressions for E namely:

*
E*_{c} = -4Gm D_{cr} r / ( D_{cr}^{2} - r^{2} )^{2} from (12e)
and,

*
E*_{c} = - 4G m r / D_{cr}^{3} from (12f)

When one replaces *r* by *α D*_{cr} and equates these expressions, one obtains

*
- 4Gm / D*_{cr}^{2} ( 1 - α ^{2} )^{2} = -4GmD_{cr} / D_{c0}^{3}
from which,

*
D*_{cr} = D_{c0} ( l - α^{2} )^{-2/3} ...... (15b)

From this one can calculate that when *r* is 1 per cent of *D*_{cr} the ratio *D*_{cr} / D_{c0} is 1.000067, and when *r* is 5 per cent of D_{cr} it is 1.00167.

If one puts for *D* in equation (15a) *D*_{c0} when *t = 0*, one can write

*
D*_{cr} = D_{c0}e^{Ht}

From this it follows that *t* is 350,000 years when the cloud extends to
I per cent of the distance to the nearest nebula, and 8.8 million years when
the cloud extends to 5 per cent.

These figures are over-estimates, and probably considerable ones, for
they do not allow for the flattening of the slope that is occasioned by the
mass of the cloud itself. When this is taken into consideration one must
arrive at shorter, and probably substantially shorter, times, particularly
for the 5 per cent distance.

The estimate does not allow, either, for any change in the mass *m* of
the neighbouring galaxies. The conclusion has been reached in the last
section that this is growing only slowly when cloud formation begins. It
will be shown in Appendix B that it may be dwindling later. For the
condition that is being considered it must be near the turning point,
and so neglect of change of mass is not likely to introduce a big
error.

It has been found herewith that, compared with the three-and-a-half
thousand million years that are available for the cloud to acquire its final
mass, the time that it needs to become rather extensive, though it is also
still very tenuous, is quite short.

The orders of magnitude in the model based on Symmetrical Impermanence seem to fit.

15.3: *The Rate of Growth of the Cloud's Domain*

As soon as a cloud has become massive enough to have an appreciable
gravitational field of its own, a reversal zone occurs just around the
astronomical summit. I have previously described this zone as the lip of a
crater. There is some incipient cloud on both sides of this. A portion
of this cloud occupies the crater and another portion lies on the outer
slopes.

As the mass of the cloud increases, this reversal zone moves outwards;
the crater grows in depth and diameter. It continues to do so until the
reversal zone coincides with the fringe of the cloud. The first stage of
growth then ends and the second stage begins.

We thus have to picture two simultaneous movements, both proceeding
radially outwards. One is the movement of the fringe of the cloud, the other
that of its reversal zone. This begins like the lip of a crater; at a later stage
it envelops the whole cloud, and it extends eventually far out into space, as
a range of passes and summits.

The reversal zone can form only after the very tenuous incipient cloud
has become fairly extensive, and even then it is no more than a small shell
right at the centre of the cloud. To extend eventually beyond the cloud
it must grow outwards more quickly than the fringe does. Some simple
mathematics shows that this happens.

In practice the cloud's shape must be very irregular, but for the sake of
simplicity a spherical shape will be assumed here. This simplification may
lead to quantities that would need a fairly large correction factor. But it
suffices to give an idea of the relative movements of the fringe and the
boundary.

Let at any moment the distance from the centre of the cloud to the
reversal zone be *D*_{1} and let the distance from the neighbouring galaxy to
the same point on the reversal zone be *D*_{2}. Let at the same moment the
mass of the cloud within its domain be *m*_{1} and the mass within the domain
of the neighbouring galaxy be *m*_{2}. By the inverse square law

*
( D*_{1} / D_{2} )^{2} = m_{1} / m_{2} ...... (15c)

Let the distance from the centre of the cloud to its fringe be *r* and let the
further simplification be introduced that the mass of the cloud is proportional to its volume. One can then write

*
m*_{1} = α r^{3} ...... (15d)

where *α* depends on the shape and average density of the cloud. Combining
these two equations gives

*
D*_{1} = D_{2} ( α / m_{2} ) ^{1/2} r^{3/2} ...... (15e)

If *D*_{2}, α and *m*_{2} have given values, *D*_{1} is seen from this equation to increase
more rapidly than *r*. This is perhaps easier to appreciate if one replaces
equation (15e) by

*
D*_{1} / r = constant x r^{1/2} ...... (15f)

This expression shows that *D*_{1} is smaller than *r* when *r* is small and larger
than *r* when *r* is large. There is one particular value for *r* when it equals *D*_{1}.
This is the value for which the fringe of the cloud coincides with the
reversal zone.

Too much importance must not be given to the above equations. They
do little more than to provide reassurance that in the model based on
Symmetrical Impermanence the time must arrive rather soon in the history
of an extragalactic cloud when it ceases to grow in extent, when indeed it
begins to shrink under its own gravitational field, and when at the same
time it grows in mass by capture of hydrogen from without.

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